Volume 5, 2001
|Page(s)||141 - 170|
|Published online||15 August 2002|
Sur quelques algorithmes récursifs pour les probabilités numériques
Laboratoire de Probabilités et
Modèles Aléatoires, UMR 7599,
Université Pierre et Marie Curie, Université Paris 6, Case 188, 4 place Jussieu,
75252 Paris Cedex 05, France ;
Revised: 17 September 2001
The aim of this paper is to take an in-depth look at the long time behaviour of some continuous time Markovian dynamical systems and at its numerical analysis. We first propose a short overview of the main ergodicity properties of time continuous homogeneous Markov processes (stability, positive recurrence). The basic tool is a Lyapunov function. Then, we investigate if these properties still hold for the time discretization of these processes, either with constant or decreasing step (ODE method in stochastic approximation, Euler scheme for diffusions). We point out several advantages of the weighted empirical random measures associated to these procedures, especially with decreasing step, in terms of convergence and of rate of convergence. Several simulations illustrate these results.
Mathematics Subject Classification: 65C30 / 62L20
Key words: Ergodicity / stability / Markov process / diffusion / stochastic algorithm / ODE method / Euler scheme / empirical measure.
© EDP Sciences, SMAI, 2001
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